Question

Which of the following function is not continuous at x = 0?

Options

• f(x) = `{{:(= (1 + 2x)^(1//x)",", x ≠ 0),(= e^2",", x = 0):}`

• f(x) `{{:(= sin x - cos x",", x ≠ 0),(= -1",", x = 0):}`

• f(x) `{{:(= (e^(1//x) - 1)/(e^(1//x) + 1)",", x ≠ 0),(= -1",", x = 0):}`

• f(x) `{{:(= (e^(5x) - e^(2x))/(sin 3x)",", x ≠ 0),(= 1",", x = 0):}`

Answer 1

`bb(f(x) {{:(= (e^(1//x) - 1)/(e^(1//x) + 1)",", x ≠ 0),(= -1",", x = 0):}`

Explanation:

Here, f(0) = – 1

and `lim_(x->0) "f"(x) = lim_(x->0) (e^(1//x) - 1)/("e"^(1//x) + 1) = lim_(x->0) (1 - e^(-1//x))/(1 + e^(-1//x))` = 1

∴ f(0) ≠ `lim_(x->0) "f"(x)`

∴ f(x) is not continuous.